Fall 2016

With W. Morris in 1992, I introduced the idea of comparing polytopes relevant to combinatorial optimization via calculation of n-dimensional volumes. I will review some of that work (related to fixed-charge problems) and describe some new work, with E. Speakman, relevant to the spatial branch-and-bound approach to global optimization. In this new work, we calculate exact expressions for 4-dimensional volumes of natural parametric families of polytopes relevant to different convex relaxations of trilinear monomials. As a consequence, we have practical guidance (i) for tuning an aspect of spatial branch-and-bound implementations, and (ii) at the modeling level.

Nov

22

Selective Migration, Occupational Choice, and the Wage Returns to College Majors

I examine the extent to which the returns to college majors are influenced by selective migration and occupational choice across locations in the US. To quantify the role of selection, I develop and estimate an extended Roy model of migration, occupational choice, and earnings where, upon completing their education, individuals choose a location in which to live and an occupation in which to work. In order to estimate this high-dimensional choice model, I make use of machine learning methods that allow for model selection and estimation simultaneously in a non-parametric setting. I find that selection in location and occupational choice can have large effects on earnings differences across majors, but that the effect is small in most locations. Using estimates of the model, I characterize the migration behavior of different college majors and find that migration flows are driven by cross-location differences in both wage returns and availability of related occupations. This finding has important implications for local governments seeking to attract or retain skilled workers.

Spectral graph is the study of eigenvalues of graphs and their connections to the graphs combinatorial properties. In this talk, I will present some of my favorite results in spectral graphs involving expanders, graph decomposition and addressings, strongly regular graphs and spectral characterization of graphs. The talk should be accessible to undergraduate students and I will present several open problems.

Nov

09

Does the threat of suspension curb dangerous behavior in soccer? A case study from the Premier League

Using data from the 2011-2012 season of the Premier League, we study
empirically and theoretically the impact of soccer suspension rules on
the behavior of players and referees. For players facing a potential
1-game suspension, being one versus two yellow cards away from the
suspension limit results in an approximate 12% reduction in fouling,
while for those facing a potential 2-game suspension, the reduction is
approximately 23%. The probability such players receive a yellow card
is also reduced. In addition, we find some evidence of slight referee
bias for the home-team in the dispensing of penalty cards, but not in
the calling of fouls. Finally, we develop a theoretical framework for
investigating the effects of suspension rules on the number of fouls
committed. Within this framework, we investigate how policy
instruments such as referees' propensity to give out yellow cards or
their consistency in giving them out affect the impact of suspension
rules.

A graded module over a polynomial ring is free if it has a set of generators so that every module element can be written uniquely as a polynomial combination of the given generators. This is the algebraic analog of a vector bundle over projective space splitting completely. I will discuss two contexts in which freeness plays a crucial role. The first, from approximation theory, is the module of piecewise polynomial functions (splines) over a fixed subdivision. The second, from the theory of hyperplane arrangements, is the module of derivations. Classical work on freeness in these contexts contains a beautiful interplay between combinatorial and geometric features, which we will spend most of the talk exploring. Towards the end we will discuss recent work on connections between splines and hyperplane arrangements.

Geometric group theory or more precisely large scale geometry of discrete groups is based on the fundamental observation that the word metrics on a discrete group given by distinct finite generating sets are bi-Lipschitz equivalent, i.e., differ at most by a multiplicative constant. This observation permits treating finitely generated groups as geometric objects as long as the methods employed are insensitive to the multiplicative error and has led to a very rich interplay between numerous mathematical disciplines such as algebra, topology, functional and harmonic analysis, ergodic theory and logic. Moreover, this study carries quite easily over to compactly generated locally compact groups, but so far more general topological transformation groups, e.g., homeomorphism and diffeomorphism groups, have resisted treatment from this perspective due to the presumed absence of canonical generating sets. We shall present some newly developed tools for overcoming this. The talk will be aimed at a general audience.

Our understanding of tropical weather and climate is less advanced than our understanding of weather and climate in the midlatitudes, where most of the United States is located. What is different about the tropics? One important difference is that clouds and rain appear not along fronts but in seemingly random clusters. As a result, one can imagine that tropical weather and climate could be modeled using stochastic partial differential equations (PDEs). In this talk, stochastic PDE models are presented for tropical rainfall and coupling with equatorial waves. Comparisons with observational data will be shown for several of the main features of tropical rainfall, such as the Madden-Julian oscillation and the distribution of cloud cluster sizes. Implications for long-range weather forecasting, for weeks or a month in advance, will also be discussed.

Multiple zeta values (MZVs) are real numbers
indexed by a string of positive integers,
defined by a nested infinite series. They
have appeared in a surprising number of ways
in mathematics and physics. A slight change
in the definition gives multiple zeta-star
values (MZSVs). Both MZVs and MZSVs satisfy
many remarkable identities. Recently S. Yamamoto
introduced interpolated multiple zeta values,
which involve a parameter r; r = 0 gives MZVs
and and r = 1 gives MZSVs. Interpolated multiple
zeta values allow common proofs of identities
for MZVs and MZSVs, and the case r = 1/2 is
worthy of study in its own right.

Identifying trends within two-dimensional data is a common challenge across the sciences, and the theory of permutation patterns adds new tools to this problem. One permutation is said to occur as a pattern in a larger one if we can find entries in the larger permutation which are in the same relative order as those of the smaller. By translating sets of points on a plane to permutations, we can use the language of permutations to describe and explore patterns. Pattern occurrences translate to topological invariants of a dataset, the statistics of which have only recently been studied.
In this talk we investigate the following question: How does the absence of one pattern affect the number of occurrences of another? This has led to several interesting and surprising identities, concerning both individual patterns and the number of patterns with the same distribution across a set of permutations. We start by exploring the notion of pattern-avoiding sets of permutations, before analyzing the number of small patterns in pattern-avoiding permutations and classifying pattern occurrence identities within the separable permutations.
This talk will be accessible to a wide audience, and will include plenty of pictures.